Force Analysis of Machinery. Introduction: In a dynamic analysis, we create equations that relate force and motion of a body (as in ME 233) or in our case a mechanism or machine. These are called equations of motion .
In a dynamic analysis, we create equations that relate force and motion of a body (as in ME 233) or in our case a mechanism or machine. These are called equations of motion.
There are 2 directions to these problems: the Forward Dynamics problem, where the motion is given and the forces are to be determined, and the Inverse Dynamics problem, where the driving force is given and the resulting motion is to be found.
Forward kinematics: We will call our method Kinetostatics – look for dynamic equilibrium at a specific position and time, a snapshot of the mechanism.
The equations will look like:
With m, a known, F unknown, or for a mechanism:
The eq’s for force are linear and solved using linear algebra
Inverse dynamics: Called the time-response problem. This solves the motion of a mechanism given the input driving force. Force example, the time history of the flight of an arrow leaving a bow.
For these problems, we write equations of motion (which are now differential eqations of motion) that might look like:
And solve motion as a function of time through numerical integration. For ex.
Given motion, find required driving force and all bearing reactions
Acceleration of pt. P:
Sum forces on particle P:
And integrate to solve:
A link is a 2FM if it satisfies 3 conditions:
All forces lie along direction of the link
Find: Tin and reactions at grd. bearings
In the matrix method, equations of dynamic equilibrium are written for FBD’s of all the links in the mechanism w/ all internal and external forces included. This results in a coupling of the unknown forces. However, the equations are linear in these forces and may be solved using linear algebra techniques.
Known motion info
Unknown forces and torques
In a general mechanism, there may be anywhere from 10 to 30 unknown forces to solve. Solve in Matlab (or other computer program)
This method will be demonstrated first on a four-bar linkage.
The figure above shows a general 4-bar linkage. The center of mass of each link is shown, as well as the input torque on link 2, and an applied torque (T4) on link 4.
Here, we have included some additional vectors to help define our problem. This leads to the following notation:
gi = center of mass of link i
jti = joint i
rij = vector from cm of i to jt. j
Fij = vector force of i on j
Solve for the input torque and all bearing reactions using the matrix method – set up the linear system of equations in matrix form.
At this joint, there are 2 unknown vector reactions, say F12 and F13
The direction of the force between the gears is known (along the common normal), with the magnitude unknown